Running w/out a gas cap or not tight enough will not affect driveability. In OBDII systems, will let the driver know something is wrong thru the CEL illuminating, registering a code, or with the newer "cap missing/loose" indicator, but the engine will continue to run with no noticeable problems.
The only way a cap would have something to do in creating a driveability problem, is if the evap system is shut down and the cap is closed tight, not allowing the vapors to be released or recycled, causing a vacuum in the tank. This will not allow the fuel to flow to the engine eventually, the stronger the vacuum in the tank, the less fuel that will be able to get out of the tank. My 2¢.